6.4.1 Derivatives of Inverse Function

Question 1

Derivatives of Inverse Functions

Answer 1

• You can calculate the derivative of an inverse function at a point without determining the actual inverse function.

Question 2

note

Answer 2

• The inverse of a function retains many of the properties of the original function.
• To derive the formula for the derivative of an inverse, start with a relationship you know.
• The composition of a function and its inverse is equal to x. You need to use implicit differentiation. Use the chain rule to differentiate both sides of that relationship.
• Isolate the derivative of the inverse by dividing.
• If you know the value of the inverse at a point, you can find the derivative of the inverse at that point.
• In this example, you know the function and the value of the inverse at π. Your mission is to find the value of the
  derivative of the inverse at π.
• Use the formula that you learned above. The derivative of f(x) = 2x + cos x is f ́(x) = 2 – sin x. Sine takes on values between –1 and 1, so the derivative lies between 1 and 3. It’s always positive, which means the function is increasing. Remember that increasing functions are invertible.
• Once you have found the derivative of the original function and verified that the function is invertible, all you have to do is plug into the formula.
• You have evaluated the derivative of the inverse of a function at a point, without determining the inverse itself!

Question 3

If f is an invertible function, which of the following is not true?

Answer 3

If f is increasing then f −1 is decreasing.

Question 4

If f (x) = (e^ x − e^ −x ) / 2 and f −1 (0) = 0, find the derivative of f −1 at x = 0.

Answer 4

1

Question 5

If f (x) = x + ln x, where x > 0, and
f −1 (1 + e) = e, find the derivative of
f −1 at x = 1 + e.

Answer 5

e/e+1

Question 6

Let f be a function. If f′(x)≥2, for any x,which of the following is true?

Answer 6

d/dx[f−1(x)]≤1/2, for any x

Question 7

If f (x) = x ^3 + 3x, and f −1 (4) = 1, find the derivative of f −1 at x = 4.

Answer 7

1/6

Question 8

If f (x) = sin x + e^ x + x and f −1 (1) = 0, find the derivative of f −1 at x = 1.

Answer 8

1/3

Question 9

If f (x) = sin^2 x − 2x, and f −1 (0) = 0, find the derivative of f −1 at x = 0.

Answer 9

-1/2

Question 10

If f (x) = x ^101 + 101x, and f −1 (102) = 1, find the derivative of f −1 at x = 102.

Answer 10

1/202

Question 11

If f(x)=sinx−3x and f−1(−3π)=π, find the derivative of f−1 at x=−3π.

Answer 11

-1/4

Question 12

If f(x)=x+e^x and f−1(1)=0, find the derivative of f−1 at x=1.

Answer 12

1/2